\(\int x (a x+b x^3)^{3/2} \, dx\) [47]

Optimal result

Integrand size = 15, antiderivative size = 304 \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=-\frac {8 a^3 x \left (a+b x^2\right )}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}+\frac {8 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}}-\frac {4 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}} \]

[Out]

2/13*x^2*(b*x^3+a*x)^(3/2)-8/65*a^3*x*(b*x^2+a)/b^(3/2)/(a^(1/2)+x*b^(1/2))/(b*x^3+a*x)^(1/2)+8/195*a^2*x*(b*x
^3+a*x)^(1/2)/b+4/39*a*x^3*(b*x^3+a*x)^(1/2)+8/65*a^(13/4)*(cos(2*arctan(b^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/co
s(2*arctan(b^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*
b^(1/2))*x^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(7/4)/(b*x^3+a*x)^(1/2)-4/65*a^(13/4)*(cos(2*arctan
(b^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x^(1
/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*x^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(7/4)/(b*x^3+
a*x)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2046, 2049, 2057, 335, 311, 226, 1210} \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=-\frac {4 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}}+\frac {8 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}}-\frac {8 a^3 x \left (a+b x^2\right )}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2} \]

[In]

Int[x*(a*x + b*x^3)^(3/2),x]

[Out]

(-8*a^3*x*(a + b*x^2))/(65*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) + (8*a^2*x*Sqrt[a*x + b*x^3])/(195
*b) + (4*a*x^3*Sqrt[a*x + b*x^3])/39 + (2*x^2*(a*x + b*x^3)^(3/2))/13 + (8*a^(13/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]
*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(65*b^(7/4)
*Sqrt[a*x + b*x^3]) - (4*a^(13/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Elli
pticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(65*b^(7/4)*Sqrt[a*x + b*x^3])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n +
1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}+\frac {1}{13} (6 a) \int x^2 \sqrt {a x+b x^3} \, dx \\ & = \frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}+\frac {1}{39} \left (4 a^2\right ) \int \frac {x^3}{\sqrt {a x+b x^3}} \, dx \\ & = \frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3\right ) \int \frac {x}{\sqrt {a x+b x^3}} \, dx}{65 b} \\ & = \frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{65 b \sqrt {a x+b x^3}} \\ & = \frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}-\frac {\left (8 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{65 b \sqrt {a x+b x^3}} \\ & = \frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}-\frac {\left (8 a^{7/2} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{65 b^{3/2} \sqrt {a x+b x^3}}+\frac {\left (8 a^{7/2} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{65 b^{3/2} \sqrt {a x+b x^3}} \\ & = -\frac {8 a^3 x \left (a+b x^2\right )}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}+\frac {8 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}}-\frac {4 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.28 \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\frac {2 x \sqrt {x \left (a+b x^2\right )} \left (\left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{13 b \sqrt {1+\frac {b x^2}{a}}} \]

[In]

Integrate[x*(a*x + b*x^3)^(3/2),x]

[Out]

(2*x*Sqrt[x*(a + b*x^2)]*((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] - a^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/
a)]))/(13*b*Sqrt[1 + (b*x^2)/a])

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {2 x^{2} \left (15 b^{2} x^{4}+25 a b \,x^{2}+4 a^{2}\right ) \left (b \,x^{2}+a \right )}{195 b \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(210\)
default	\(\frac {2 b \,x^{5} \sqrt {b \,x^{3}+a x}}{13}+\frac {10 a \,x^{3} \sqrt {b \,x^{3}+a x}}{39}+\frac {8 a^{2} x \sqrt {b \,x^{3}+a x}}{195 b}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(217\)
elliptic	\(\frac {2 b \,x^{5} \sqrt {b \,x^{3}+a x}}{13}+\frac {10 a \,x^{3} \sqrt {b \,x^{3}+a x}}{39}+\frac {8 a^{2} x \sqrt {b \,x^{3}+a x}}{195 b}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(217\)

[In]

int(x*(b*x^3+a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/195*x^2*(15*b^2*x^4+25*a*b*x^2+4*a^2)/b*(b*x^2+a)/(x*(b*x^2+a))^(1/2)-4/65/b^2*a^3*(-a*b)^(1/2)*((x+(-a*b)^(
1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*x^3+a*
x)^(1/2)*(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^(1/2)/b*El
lipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.22 \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\frac {2 \, {\left (12 \, a^{3} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (15 \, b^{3} x^{5} + 25 \, a b^{2} x^{3} + 4 \, a^{2} b x\right )} \sqrt {b x^{3} + a x}\right )}}{195 \, b^{2}} \]

[In]

integrate(x*(b*x^3+a*x)^(3/2),x, algorithm="fricas")

[Out]

2/195*(12*a^3*sqrt(b)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + (15*b^3*x^5 + 25*a*b^2*x
^3 + 4*a^2*b*x)*sqrt(b*x^3 + a*x))/b^2

Sympy [F]

\[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\int x \left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}\, dx \]

[In]

integrate(x*(b*x**3+a*x)**(3/2),x)

[Out]

Integral(x*(x*(a + b*x**2))**(3/2), x)

Maxima [F]

\[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\int { {\left (b x^{3} + a x\right )}^{\frac {3}{2}} x \,d x } \]

[In]

integrate(x*(b*x^3+a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a*x)^(3/2)*x, x)

Giac [F]

\[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\int { {\left (b x^{3} + a x\right )}^{\frac {3}{2}} x \,d x } \]

[In]

integrate(x*(b*x^3+a*x)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^3 + a*x)^(3/2)*x, x)

Mupad [F(-1)]

Timed out. \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\int x\,{\left (b\,x^3+a\,x\right )}^{3/2} \,d x \]

[In]

int(x*(a*x + b*x^3)^(3/2),x)

[Out]

int(x*(a*x + b*x^3)^(3/2), x)